I think this is something I should have known, but if I ever did I forgot about it. Consider the field $L$ of $p$-th roots of unity ($p$ prime) and its maximal real subfield $L^+$. The transfer of ideal classes $\iota: Cl(L^+) \longrightarrow Cl(L)$ is injective, hence the exact sequence $$ 1 \longrightarrow Cl(L^+) \longrightarrow Cl(L) \longrightarrow Cl(L)^- \longrightarrow1 $$ defines a factor group $Cl(L)^-$ on which complex conjugation acts as $-1$: The minus class group.
On the other hand, $L/L^+$ is ramified, hence the norm map on clsas groups is surjective, and we have an exact sequence $$ 1 \longrightarrow Cl(L/L^+) \longrightarrow Cl(L) \longrightarrow Cl(L^+) \longrightarrow 1. $$ The relative class group $Cl(L/L+)$ clearly has the same order as $Cl(L)^-$, and my question is whether they are isomorphic.
The natural map sending $c \in Cl(L/L^+)$ to the coset $c \cdot Cl(L^+) \in Cl(L)^-$ has kernel $Cl(L/L^+)[2]$, hence the odd parts of $Cl(L/L+)$ and $Cl(L)^-$ are in fact isomorphic. But for the $2$-Sylow subgroups this is not so clear. In fact I almost expect that there should be a counterexample, probably coming from cubic subfields with large $2$-rank, but so far I have not been able to pin one down. Is there a known example with non-isomorphic $2$-parts, or am I hunting a ghost?
